Derived invariance of the cap product in Hochschild theory
aa r X i v : . [ m a t h . K T ] N ov DERIVED INVARIANCE OF THE CAP PRODUCT INHOCHSCHILD THEORY
MARCO A. ARMENTA A. AND BERNHARD KELLER
Abstract.
We prove derived invariance of the cap product for associa-tive algebras projective over a commutative ring. Introduction
It has been known since Rickard’s work [3] that Hochschild cohomologyis preserved under derived equivalence as a graded algebra with the cupproduct. Using the methods of [3], one can also show that Hochschild ho-mology is preserved as a graded space, see for example [5]. Nevertheless,derived invariance of the cap product – which provides an action of theHochschild cohomology algebra on the Hochschild homology – has not beenconsidered. In this note, we prove that derived invariance holds as well forthe cap product.This paper is part of the Ph. D. thesis of the first author, whose advisorsare Claude Cibils and Jos´e Antonio de la Pe˜na, to whom he is very grateful.It enters into the first author’s project of showing the derived invariance ofthe Tamarkin-Tsygan calculus associated with a k -projective algebra.2. Derived invariance
Let k be a commutative ring and A an associative k -algebra, projective asa k -module. We write A e for the envelopping algebra A ⊗ k A op . We denoteby D ( A ) the unbounded derived category of the category of right A -modules.For a bimodule M , we denote by HH • ( A, M ) the Hochschild cohomologywith coefficients in M and by HH • ( A, M ) the Hochschild homology withcoefficients in M , see for example [1] or [4]. We have canonical isomorphisms HH n ( A, M ) ∼ → H n (RHom A e ( A, M )) = Hom D ( A e ) ( A, M [ n ])and HH n ( A, M ) ∼ → H n ( A L ⊗ A e M ) . Let f ∈ HH m ( A, A ). The cap product by f is a map f ∩ ? : HH n ( A, M ) → HH n − m ( A, M ) . Key words and phrases.
Derived category, Hochschild homology, Hochschildcohomology.
The following lemma gives an interpretation of the cap product in terms ofthe derived category
Lemma 2.1.
The following square commutes, where the vertical arrows arethe canonical identifications. HH m ( A, M ) (cid:15) (cid:15) f ∩ ? / / HH m − n ( A, M ) (cid:15) (cid:15) H ( M L ⊗ A e A [ − m ]) H (id ⊗ f ) / / H ( M L ⊗ A e A [ n − m ]) . Proof.
Let
Bar ( A ) be the bar resolution of A , we get M L ⊗ A e A = T ot ( M ⊗ A e Bar ( A )) = M ⊗ A e Bar ( A ) . Let x ∈ M and y ∈ Bar ( A ), then H (id ⊗ f )([ x ⊗ y ]) = [ x ⊗ f ( y )] = f ∩ [ x ⊗ y ] . (cid:3) Now suppose that A is derived equivalent to a k -projective algebra B . ByRickard’s Morita theory for derived categories [2] [3], this implies that thereexist bimodule complexes X ∈ D ( A op ⊗ k B ) and X ∨ ∈ D ( B op ⊗ k A ) such thatthere are isomorphisms η : A ∼ → X L ⊗ B X ∨ and ε : X ∨ L ⊗ A X ∼ → B in D ( A e )respectively D ( B e ). We may and will suppose that these isomorphisms makethe following triangles commutative: X η ⊗ X / / = & & ▼▼▼▼▼▼▼▼▼▼▼▼▼▼ X L ⊗ B X ∨ L ⊗ A X X ⊗ ε (cid:15) (cid:15) X X ∨ X ∨ ⊗ η / / = & & ◆◆◆◆◆◆◆◆◆◆◆◆◆ X ∨ L ⊗ A X L ⊗ B X ∨ ε ⊗ X ∨ (cid:15) (cid:15) X ∨ . As a consequence, the functor F =? L ⊗ A e ( X L ⊗ k X ∨ ) : D ( A e ) → D ( B e )is an equivalence with quasi-inverse G =? L ⊗ B e ( X L ⊗ k X ∨ ). We have canonicalisomorphisms F A = A L ⊗ A e ( X L ⊗ k X ∨ ) = X ∨ L ⊗ A A L ⊗ A X = X ∨ L ⊗ A X ∼ → B and GB = B L ⊗ B e ( X ∨ L ⊗ k X ) = X L ⊗ B B L ⊗ B X ∨ = X L ⊗ B X ∨ ∼ → A. We obtain a canonical isomorphism HH n ( A, A ) = Hom D ( A e ) ( A, A [ n ]) ∼ → Hom D ( B e ) ( X ∨ L ⊗ A X, X ∨ L ⊗ A X [ n ]) ∼ → Hom D ( B e ) ( B, B [ n ]) = HH n ( B, B ) . ERIVED INVARIANCE OF THE CAP PRODUCT 3
By abuse of notation, we will still denote it by f F f . Let us supposethat M is an A -bimodule such that N = F M is concentrated in degree 0.For example, if M = A , then N = B . Theorem 2.2.
There is a canonical isomorphism HH • ( A, M ) ∼ → HH • ( B, N ) such that for each f ∈ HH m ( A, A ) the following square commutes HH n ( A, M ) f ∩ ? / / ∼ = (cid:15) (cid:15) HH n − m ( A, M ) ∼ = (cid:15) (cid:15) HH n ( B, N ) F f ∩ ? / / HH n − m ( B, N ) . Proof.
We define the isomorphism HH • ( A, M ) ∼ → HH • ( B, N )to be induced by the canonical chain of isomorphisms in D ( k ) M L ⊗ A e A ∼ → M L ⊗ A e ( X L ⊗ B X ∨ ) = M L ⊗ A e ( X L ⊗ k X ∨ ) L ⊗ B e B = F M L ⊗ B e B = N L ⊗ B e B. Let f ∈ HH m ( A, A ). It suffices to show that the following square is com-mutative M L ⊗ A e A / / M ⊗ f (cid:15) (cid:15) M L ⊗ A e ( X L ⊗ k X ∨ ) L ⊗ B e B M ⊗ X ⊗ X ∨ ⊗ F f (cid:15) (cid:15) M L ⊗ A e A [ m ] / / M L ⊗ A e ( X L ⊗ k X ∨ ) L ⊗ B e B [ m ] . This is implied by the commutativity of the square A f (cid:15) (cid:15) / / B L ⊗ B e ( X ∨ L ⊗ k X ) ( F f ) ⊗ X ∨ ⊗ X (cid:15) (cid:15) A [ m ] / / B [ m ] L ⊗ B e ( X ∨ L ⊗ k X ) . In turn, this will follow from the commutativity of the square A f (cid:15) (cid:15) / / A L ⊗ A e ( X L ⊗ k X ∨ ) L ⊗ B e ( X ∨ L ⊗ k X ) f ⊗ X ⊗ X ∨ ⊗ X ∨ ⊗ X (cid:15) (cid:15) A [ m ] / / A [ m ] L ⊗ A e ( X L ⊗ k X ∨ ) L ⊗ B e ( X ∨ L ⊗ k X ) . This last commutativity follows from the naturality of the adjunction mor-phism A ∼ → GF A . (cid:3) MARCO A. ARMENTA A. AND BERNHARD KELLER
References
1. Henri Cartan and Samuel Eilenberg,
Homological algebra , Princeton Landmarks inMathematics, Princeton University Press, Princeton, NJ, 1999, With an appendix byDavid A. Buchsbaum, Reprint of the 1956 original.2. Jeremy Rickard,
Morita theory for derived categories , J. London Math. Soc. (2) (1989), no. 3, 436–456.3. , Derived equivalences as derived functors , J. London Math. Soc. (2) (1991),no. 1, 37–48.4. Charles A. Weibel, An introduction to homological algebra , Cambridge Studies in Ad-vanced Mathematics, vol. 38, Cambridge University Press, Cambridge, 1994.5. Alexander Zimmermann,
Fine Hochschild invariants of derived categories for symmet-ric algebras , J. Algebra (2007), no. 1, 350–367.
M. A. : Centro de Investigaci´on en Matem´aticas A. C., Cubicle D104, 36240Guanajuato, Gto. M´exicoB. K. : Universit´e Paris Diderot – Paris 7, UFR de Math´ematiques, Institutde Math´ematiques de Jussieu–PRG, UMR 7586 du CNRS, Case 7012, BˆatimentSophie Germain, 75205 Paris Cedex 13, France
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